Denote $I_k^l=[k\cdot3^l,(k+1)\cdot3^l)\cap\mathbb{Z}$, call a set $A\subset\mathbb{Z}$ "$l$-good" if it satisfy the condition inthe question with $2023\to l$. Lemma 1: Suppose $A\subset I_k^l\subset I_{k'}^{l+1}$ and $B\subset I_t^l\subset I_{k'}^{l+1}$ are $l$-good sets $(t\neq k)$, then $A\cup B\subset I_{k'}^{l+1}$ is an $l+1$-good set. Lemma 2: Suppose $S\subset I_k^l$, then one of the two following claims must hold: 1) There exists an $l$-good set $A\subset I_k^l$ such that $A\subset S.$ 2) There exists an $l$-good set $B\subset I_k^l$ such that $B\cap S=\emptyset.$ (induct on $l$) Consider $S\cap I_0^{2023}\subset I_0^{2023}$ in lemma 2 kills the problem.